Remark 1 
(Selection of \(\lambda \) ) On the one hand, the
regularization parameter \(\lambda \)  needs to be large enough for the
regularized OT problem to be close to the original OT problem. This
is indeed reflected in the choice \(\lambda = 4 \, \varepsilon ^{-1}\log n\)  that enables achieving an \(\varepsilon \) -approximation of the
optimal transport plan: the smaller the tolerance
\(\varepsilon \) , the larger \(\lambda \) . On the other hands, the
convergence of Sinkhorn's algorithm deteriorates as \(\lambda \rightarrow \infty \) ;
see e.g. [1]}, [2]}. Moreover, as \(\lambda \rightarrow \infty \) , more and more entries of the kernel \(Q\)  (and
hence entries of \(Q \, {\bf v}\)  and \(Q^{\top } {\bf u}\) ) may
become “essentially” zero. In this case, Sinkhorn's algorithm may become numerically
unstable due to division by zero. The selection of the regularization parameter
\(\lambda \)  should therefore be based on a trade-off between
accuracy, computational cost, and robustness.

